Subgroups — C₂³×#88142 #16769609

Order 3328.

12190 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₆	C₂	1	2	yes	maximal
2	C₂²×#88142	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#6101	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
10	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₂₆	—	2	32	no	—
13	C₂³×Dic₂₆	C₂²	1	56	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	28	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	—
16	C₂²×#6101	C₂²	1	56	yes	—
17	C₂²×C₄×Dic₁₃	C₂²	1	28	yes	—
18	C₂×#88142	C₂²	1	448	yes	direct product factor
19	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	—	2	16	no	—
21	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
22	C₂³×C₄×C₁₃	C₂³	1	28	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
24	C₂³×Dic₁₃	—	2	16	no	—
25	C₂³×Dic₁₃	D₄	1	16	yes	—
26	C₂³×Dic₁₃	Q₈	1	16	yes	complemented
27	C₂³×Dic₁₃	C₂³	1	28	yes	—
28	C₂³×Dic₁₃	C₂³	1	14	yes	—
29	C₂³×Dic₁₃	C₂³	1	14	yes	—
30	C₂²×Dic₂₆	—	2	448	no	—
31	C₂²×Dic₂₆	C₂³	1	224	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	—
33	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	—
34	C₂×#6101	C₂³	1	224	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	112	yes	—
36	#88142	C₂³	1	512	yes	direct product factor
37	C₂³×C₄⋊Q₈	—	13	1	no	maximal, Sylow
38	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
39	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
40	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
41	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
42	C₂²×C₄×C₁₃	—	2	112	no	—
43	C₂²×C₄×C₁₃	C₂×Q₈	1	112	yes	complemented
44	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	—
45	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
46	C₂²×Dic₁₃	—	2	112	no	—
47	C₂²×Dic₁₃	C₂×D₄	1	112	yes	—
48	C₂²×Dic₁₃	C₂×Q₈	1	112	yes	complemented
49	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
50	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
51	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
52	C₂×Dic₂₆	—	2	896	no	—
53	C₂×Dic₂₆	C₂⁴	1	128	yes	—
54	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
55	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
56	#6101	C₂⁴	1	128	yes	—
57	C₄×Dic₁₃	C₂⁴	1	64	yes	—
58	C₂⁴×Q₈	—	13	2	no	—
59	C₂²×C₄⋊Q₈	—	13	56	no	—
60	C₂³×C₄⋊C₄	—	13	2	no	—
61	C₂³×C₄⋊C₄	—	13	1	no	—
62	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
63	C₂³×C₄²	—	13	1	no	—
64	C₂³×C₁₃	C₄⋊Q₈	1	64	yes	complemented
65	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
66	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
67	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
68	C₂³×C₁₃	C₂⁵	1	7	yes	—
69	C₂×C₄×C₁₃	—	2	112	no	—
70	C₂×C₄×C₁₃	C₂²×Q₈	1	112	yes	complemented
71	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
72	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
73	C₂×Dic₁₃	—	2	112	no	—
74	C₂×Dic₁₃	C₂²×D₄	1	112	yes	—
75	C₂×Dic₁₃	C₂²×Q₈	1	112	yes	complemented
76	C₂×Dic₁₃	C₂⁵	1	16	yes	—
77	C₂×Dic₁₃	C₂⁵	1	8	yes	—
78	C₂×Dic₁₃	C₂⁵	1	8	yes	—
79	Dic₂₆	—	2	256	no	—
80	C₂³×Q₈	—	13	56	no	—
81	C₂³×Q₈	—	26	32	no	—
82	C₂⁴×C₄	—	13	1	no	—
83	C₂⁴×C₄	—	13	1	no	—
84	C₂⁴×C₄	—	13	2	no	—
85	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
86	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
87	C₂×C₄⋊Q₈	—	13	448	no	—
88	C₂²×C₄⋊C₄	—	13	56	no	—
89	C₂²×C₄⋊C₄	—	13	28	no	—
90	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	—
91	C₂²×C₄²	—	13	28	no	—
92	C₂²×C₁₃	C₂×C₄⋊Q₈	1	112	yes	complemented
93	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
94	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
95	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
96	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
97	C₄×C₁₃	—	2	16	no	—
98	C₄×C₁₃	C₂³×Q₈	1	16	yes	complemented
99	Dic₁₃	—	2	16	no	—
100	Dic₁₃	C₂³×D₄	1	16	yes	—
101	Dic₁₃	C₂³×Q₈	1	16	yes	complemented
102	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
103	C₂²×Q₈	—	13	224	no	—
104	C₂²×Q₈	—	26	448	no	—
105	C₂³×C₄	—	2	16	no	—
106	C₂³×C₄	—	13	16	no	—
107	C₂³×C₄	—	13	16	no	—
108	C₂³×C₄	—	13	14	no	—
109	C₂³×C₄	—	13	14	no	—
110	C₂³×C₄	—	13	28	no	—
111	C₂³×C₄	—	26	16	no	—
112	C₂³×C₄	Dic₂₆	1	16	yes	complemented
113	C₂³×C₄	C₂²×D₁₃	1	28	yes	—
114	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
115	C₄⋊Q₈	—	13	512	no	—
116	C₂×C₄⋊C₄	—	13	224	no	—
117	C₂×C₄⋊C₄	—	13	112	no	—
118	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	—
119	C₂×C₄²	—	13	112	no	—
120	C₂×C₁₃	C₂²×C₄⋊Q₈	1	28	yes	complemented
121	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
122	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
123	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₂₆	1	8	yes	—
125	C₂⁴	D₄×D₁₃	1	8	yes	—
126	C₂⁴	Q₈×D₁₃	1	8	yes	—
127	C₂⁴	C₂³×D₁₃	1	7	yes	—
128	C₂×Q₈	—	13	128	no	—
129	C₂×Q₈	—	26	896	no	—
130	C₂²×C₄	—	2	112	no	—
131	C₂²×C₄	—	13	112	no	—
132	C₂²×C₄	—	13	112	no	—
133	C₂²×C₄	—	13	28	no	—
134	C₂²×C₄	—	13	28	no	—
135	C₂²×C₄	—	13	56	no	—
136	C₂²×C₄	—	26	112	no	—
137	C₂²×C₄	C₂×Dic₂₆	1	112	yes	complemented
138	C₂²×C₄	C₂³×D₁₃	1	56	yes	—
139	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
140	C₄⋊C₄	—	13	128	no	—
141	C₄⋊C₄	—	13	64	no	—
142	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
143	C₄²	—	13	64	no	—
144	C₁₃	C₂³×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
145	C₂³	#88142	1	64	yes	direct product factor
146	C₂³	C₂²×Dic₂₆	1	28	yes	—
147	C₂³	C₂×D₄×D₁₃	1	28	yes	—
148	C₂³	C₂×Q₈×D₁₃	1	28	yes	—
149	C₂³	C₂⁴×D₁₃	1	7	yes	—
150	Q₈	—	26	256	no	—
151	C₂×C₄	—	2	112	no	—
152	C₂×C₄	—	13	112	no	—
153	C₂×C₄	—	13	112	no	—
154	C₂×C₄	—	13	8	no	—
155	C₂×C₄	—	13	8	no	—
156	C₂×C₄	—	13	16	no	—
157	C₂×C₄	—	26	112	no	—
158	C₂×C₄	C₂²×Dic₂₆	1	112	yes	complemented
159	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
160	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
161	C₂²	C₂×#88142	1	112	yes	direct product factor
162	C₂²	C₂³×Dic₂₆	1	14	yes	—
163	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
164	C₂²	C₂²×Q₈×D₁₃	1	14	yes	—
165	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
166	C₄	—	2	16	no	—
167	C₄	—	13	16	no	—
168	C₄	—	13	16	no	—
169	C₄	—	26	16	no	—
170	C₄	C₂³×Dic₂₆	1	16	yes	complemented
171	C₂	C₂²×#88142	1	28	yes	direct product factor
172	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
173	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
174	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×#88142	C₂	direct (N × H)
18. C₂×#88142	C₂²	direct (N × H)
21. C₂³×C₄×C₁₃	Q₈	semidirect
26. C₂³×Dic₁₃	Q₈	semidirect
36. #88142	C₂³	direct (N × H)
43. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
48. C₂²×Dic₁₃	C₂×Q₈	semidirect
64. C₂³×C₁₃	C₄⋊Q₈	semidirect
70. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
75. C₂×Dic₁₃	C₂²×Q₈	semidirect
92. C₂²×C₁₃	C₂×C₄⋊Q₈	semidirect
98. C₄×C₁₃	C₂³×Q₈	semidirect
101. Dic₁₃	C₂³×Q₈	semidirect
112. C₂³×C₄	Dic₂₆	semidirect
120. C₂×C₁₃	C₂²×C₄⋊Q₈	semidirect
137. C₂²×C₄	C₂×Dic₂₆	semidirect
144. C₁₃	C₂³×C₄⋊Q₈	semidirect
145. C₂³	#88142	direct (N × H)
158. C₂×C₄	C₂²×Dic₂₆	semidirect
161. C₂²	C₂×#88142	direct (N × H)
170. C₄	C₂³×Dic₂₆	semidirect
171. C₂	C₂²×#88142	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Dic₂₆	2
C₂²×#88142	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×C₄⋊Q₈	13

Subgroups — C23×#88142 #16769609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88142 #16769609